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1 IJESRT ITERATIOAL JOURAL OF EGIEERIG SCIECES & RESEARCH TECHOLOGY SOLUTIO FOR FUZZY DIFFERETIAL EQUATIOS USIG FOURTH ORDER RUGE-KUTTA METHOD WITH EMBEDDED HARMOIC MEA DPaul Dhayabaa * JChisty Kigsto * Associate Pofesso & Picipal PG Reseach Depatmet of Mathematics Bishop Hebe College (Autoomous) Tiuchiappalli Idia Assistat Pofesso PG Reseach Depatmet of Mathematics Bishop Hebe College (Autoomous) Tiuchiappalli Idia ABSTRACT I this pape a attempt has bee made to detemie a umeical solutio fo the fist ode fuzzy diffeetial equatios by usig fouth ode Ruge-kutta embedded hamoic mea The accuacy applicability of the poposed method is illustated by solvig a fuzzy iitial value poblem with tiagula fuzzy umbe KEYWORDS: Fuzzy Diffeetial Equatios Ruge-kutta fouth ode method Embedded Hamoic Mea Tiagula Fuzzy umbe ITRODUCTIO The theoy of fuzzy diffeetial equatio plays a impotat ole i modellig of sciece egieeig poblems because this theoy epesets a atual way to model dyamical systems ude ucetaitythe applicability of the fuzzy diffeetial equatio leads to a seveal umbe of eseach woks i the ope liteatue Fist ode liea fuzzy diffeetial equatio is oe of the simplest fuzzy diffeetial equatio which appea i may applicatios Some of the eviewed eseach papes ae cited below fo bette udestig of the peset papethe cocept of fuzzy deivative was fist itoduced by SLChag LAZadeh i [6] DDubois Pade [7] discussed diffeetiatio with fuzzy featues MLpui DARalescu [24] RGoetschel WVoxma [1] cotibuted towads the diffeetial of fuzzy fuctiosthe fuzzy diffeetial equatio iitial value poblems wee extesively studied by OKaleva [1516] by SSeikkala [25]Recetly may eseach papes ae focused o umeical solutio of fuzzy iitial value poblems(fivps) umeical Solutio of fuzzy diffeetial equatio has bee itoduced by MMa MFiedma AKel [18] though Eule method by SAbbasby TAllahvialoo [1] by taylo methodruge Kutta methods have also bee studied by authos [222] VimalaSaveethaSChethupiya discussed o umeical solutio of fuzzy diffeetial equatio by uge-kutta method with highe ode deivative appoximatios [21] RGethsi Shamila ECHey Amithaaj discussed o umeical Solutios of fist ode fuzzy iitial value poblems by o-liea tapezoidal fomulae based o vaiety of Meas[13]A ew Fouth ode uge-kutta method with embedded hamoic mea fo iitial value poblems was poposed by azeeudi Yaacob Bahom Saugi[19] also it was studied by RPoalagusamy SSethilkuma[23]Followed by the itoductio this pape is ogaized as follows:i sectio 2 some basic esults of fuzzy umbes defiitios of fuzzy deivative ae give I sectio 3 the fuzzy iitial value poblem has bee discussed Sectio 4 descibes the geeal stuctue of the fouth ode uge-kutta embedded hamoic mea methodi sectio 5 the uge kutta fouth ode embedded hamoic mea i paticula fo solvig fuzzy iitial value poblem has bee discussed Fially the applicability of the method is demostated by detemiig the umeical solutio of the poblem by applyig the poposed method [36]

2 PRELIMIARIES Defiitio:(FUZZY UMBER) A abitay fuzzy umbe is epeseted by a odeed pai of fuctios ( ( ) ( )) which satisfy the followig coditios i) () ii) is a bouded ight cotiuous o-deceasig fuctio ove u fo all the the -level set is \ ( ) }; u is a bouded left cotiuous o-deceasig fuctio ove u () 1 u u fo all 1 with espect to ay 1 with espect to ay iii) ( u( ) ( )) u { x u x 1 Clealyu { x \ u( x) } is compact which is a closed bouded iteval we deote by u ( u( ) u( )) 1 Defiitio: (TRIAGULAR FUZZY UMBER) u u u A tiagula fuzzy umbe u is a fuzzy set i E that is chaacteized by a odeed tiple 3 such that u u ; u u u u l c l The membeship fuctio of the tiagula fuzzy umbe u is give by x ul ; ul x uc uc ul u( x) 1 ; x uc u x ; uc x u u uc we have : (1) u if u l (2) u if u l (3) u if u c (4) u if u c u l u c l c with Defiitio: ( - Level Set) Let I be the eal iteval A mappig y : I E is called a fuzzy pocess its - level Set is deoted by y( t) [ y( t; ) y( t; ) ] t I 1 Defiitio: (Seikkala Deivative) The Seikkala deivative '( ) y t of a fuzzy pocess is defied by [ '( 1povided that this equatio defies a fuzzy umbe as i [25] Lemma: If the sequece of o-egative umbe give positive costats A B the m y '( t) y t; ) y '( t; ) ] t I R W satisfy W 1 A W B 1 fo the A 1 W A W B A 1 [37]

3 Lemma If the sequece of o-egative umbes satisfy W 1 W A max{ W V } B V 1 V A max{ W V } B fo the give positive costats A B the U W V Lemma Let F( t u v) we have W A 1 U A U B A 1 G t u v belog to The fo abitaily fixed 1 m ' ( ) C R F V whee A 1 2A B 2B the patial deivatives of F G be bouded ove D y t y ( t ) h L C) 2 ( ( 1) 1 ) (1 2 C Max ( ; ) ( 1; ) 1 deivatives of F G G t y t y t Theoem Let F( t u v) G( t u v) belog to ' ( ) C R F [38] R F whee L is a boud of patial the patial deivatives of F G be bouded ove The fo abitaily fixed 1 the umeical solutios of y( t 1; ) y( t 1; ) Y( t ; ) solutios Y( t 1; ) 1 uifomly i t Theoem Let F( t u v) G( t u v) belog to R F covege to the exact C ' ( RF ) the patial deivatives of F G be bouded ove RF 2Lh 1The fo abitaily fixed 1the iteative umeical solutios of ( j) y ( t ; ) covege to the umeical solutios y( t; ) y( t; ) i t t t whe j ( j y ) ( t ; ) FUZZY IITIAL VALUE PROBLEM Coside a fist-ode fuzzy iitial value poblem y( t) f ( t y( t)) t t T y( t) y whee y is a fuzzy fuctio of t f ( t y ) is a fuzzy fuctio of the cisp vaiable ' t ' the fuzzy vaiable y the fuzzy deivative of y y( t ) y We deote the fuzzy fuctio y by y y [ y( t) ] [ y( t ; ) y( t; ) ] yt [ y( t; ) y( t; )] ( 1] ( ) we wite f ( t y) [ f ( ) f( t y) ] ty f ( t y) F[ t y y ] f ( t y) G [ t y y] because of y f ( t y ) we have (31) y ' is is a tiagula o a tiagula shaped fuzzy umbe [ y] It meas that the -level set of () yt fo t t T is f t y( t); F[ t y( t ; ) yt ( ; ) ] (32) f t y( t); G[ t y( t ; ) y( t; ) ] (33)

4 by usig the extesio piciple we have the membeship fuctio f ( t y( t) )( s ) sup{ y( t)( ) \ s f ( t )} (34) so the fuzzy umbe follows that f ( t y( t) ) [ f ( t y( t); ) f ( t y( t); ) ] (35) whee ( ( ); ) mi{ f ( t u) u y( t ) } (36) f ( t y( t)) f t y t f t y( t); max{ f t u uy t Defiitio 31 A fuctio f : R RF such that t to ( ) ( ) } s R ( 1] is said to be fuzzy cotiuous fuctio if fo a abitay fixed t R (37) D f ( t) f ( t ) exists The fuzzy fuctio cosideed ae cotiuous i metic D the cotiuity of f ( t y( t); ) guaatees the existece of the defiitio of f ( t y( t); ) fo t t T 1 [1] Theefoe the fuctios G F ca be defiite too FOURTH ORDER RUGE KUTTA METHOD WITH EMBEDDED HARMOIC MEA The Fouth ode Ruge-kutta method with Embedded Hamoic Mea is a Ruge-kutta method fo appoximatig the solutio of the iitial value poblem y( t) f ( t y( t)) y( t ) y The basis of all Ruge-Kutta methods is to expess the diffeece betwee the value of y at m y y w k (41) 1 i i i whee w ' s i ae costat fo all i i1 k hf ( t a h y c k ) i i ij j j1 t 1 t (42) Iceasig the ode of accuacy of the Ruge-Kutta methodsit has bee accomplished by iceasig the umbe of Taylo s seies tems used thus the umbe of fuctioal evaluatios equied[5]the method poposed by GoekeD JohsoO[9] itoduces ew tems ivolvig highe ode deivatives of f i the Ruge-Kutta tems (i>) to obtai a highe ode of accuacy without a coespodig icease i evaluatios of f but with the additio of evaluatios of The fouth ode Ruge-kutta method with embedded hamoic mea fo step +1 which was poposed by azeeudi Yaacob Bahom Saugi[19] f ' k2 k3 2 k1k 2 2 k3k4 y( t 1) y( t) h k k 3 k k k1 hf t y( t) Whee ( ) k hf t a h y t a hk (45) k hf t ( a a ) h y( t ) a hk a hk k hf t ( a a a ) h y( t ) a hk a hk a hk (47) a a a a a a ae chose to make y 1 close to y( t 1 ) The value of paametes ae the paametes a1 a2 a3 a4 a5 a as k i (43) (44) (46) [39]

5 FOURTH ORDER RUGE-KUTTA METHOD WITH EMBEDDED HARMOIC MEA FOR SOLVIG FUZZY DIFFERETIAL EQUATIOS Let the exact solutio [ Y( t) ] [ Y( t; ) Y( t; )] is appoximated by some [ y( t)] [ y( t; ) y( t; )] The gid poits at which the solutios is calculated ae T t h t t ih i ; i Fom 43 to 47 we defie k2( t y( t )) k3( t y( t )) 2 k1( t y( t )) k2( t y( t )) k1( t y( t )) k2( t y( t )) (51) y( t 1 ) y( t ) h 2 k3( t y( t )) k4( t y( t )) 3 k3( t y( t )) k4( t y( t )) (52) whee k1 hf[ t y( t ) y( t )] h h h k2 hf[ t y( t ) k1( t y( t )) y( t ) k1( t y( t ))] (53) h h 5h k3 hf[ t y( t ) k1( t y( t )) k2( t y( t )) h 5h y( t ) k1( t y( t )) k2( t y( t ))] (54) 8 8 h 7h 9h k4 hf[ t h y( t ) k1( t y( t )) k2( t y( t )) k3( t y( t )) h 7h 9h y( t ) k1( t y( t )) k2( t y( t )) k3( t y( t ))] (55) k2( t y( t )) k3( t y( t )) 2 k1( t y( t )) k2( t y( t )) k1( t y( t )) k2( t y( t )) y( t 1 ) y( t ) h (56) 2 k3( t y( t )) k4( t y( t )) 3 k3( t y( t )) k4( t y( t )) Whee k1 hg[ t y( t ) y( t )] (57) h h h k2 hg[ t y( t ) k1( t y( t )) y( t ) k1( t y( t ))] (58) h h 5h k3 hg[ t y( t ) k1( t y( t )) k2( t y( t )) h 5h y( t ) k1( t y( t )) k2( t y( t ))] (59) 8 8 [31]

6 h 7h 9h k4 hg[ t h y( t ) k1( t y( t )) k2( t y( t )) k3( t y( t )) h 7h 9h y( t ) k1( t y( t )) k2( t y( t )) k3( t y( t ))] (51) k2( t y( t )) k3( t y( t )) 2 k1( t y( t )) k2( t y( t )) k1( t y( t )) k2( t y( t )) we defie F t y( t ) h 2 k3( t y( t )) k4( t y( t )) 3 k3( t y( t )) k4( t y( t ) ) (511) k2( t y( t )) k3( t y( t )) 2 k1( t y( t )) k2( t y( t )) k1( t y( t )) k2( t y( t )) Gt y( t ) h (512) 2 k3( t y( t )) k4( t y( t )) 3 k3( t y( t )) k4( t y( t ) ) Theefoe we have Y ( t ) ( ) [ ( )] 1 Y t F t Y t ] Y ( t ) ( ) [ ( )] 1 Y t G t Y t (513) Ad y ( t ) ( ) [ ( )] 1 y t F t y t ] (514) y ( t ) ( ) [ ( )] 1 y t G t y t Clealy y( t ; ) y( t ; ) covege to Y( t ; ) Y( t ; ) wheeve h UMERICAL EXAMPLE Coside fuzzy iitial value poblem y( t) y( t) t y() ( ) (61) The exact solutio is give by t t Y( t ) [(96 4 ) e (11 1 ) e ] At t 1 we get Y(1 ) [(96 4 ) e(11 1 ) e] 1 The values of exact appoximate solutio with h= 1 is give i Table : I The exact appoximate solutios obtaied by the poposed method is plotted i Fig:1 the eo1 eo2 is plotted i Fig:2 R Exact Solutio t=1 Table:1 Exact Appoximate Solutio Appoximate Solutio (h=1) Y( t ) Y( t ) y( t ) y( t ) Eo 1 Eo e e-6 [311]

7 e e e e e e e e e e e e e e e e e e e e-6 Figue:1 Figue:2 Exact & Appoximate Solutios Eo 1 Eo 2 [312]

8 COCLUSIO I this pape the fouth ode Ruge-Kutta method with embedded hamoic mea has beig applied fo fidig the umeical solutio of fist ode fuzzy diffeetial equatios usig tiagula fuzzy umbethe efficiecy the accuacy of the poposed method have bee illustated by a suitable example Fom the umeical example it has bee obseved that the disceet solutios by the poposed method almost coicide with the exact solutios ACKOWLEDGEMET I humbly ackowledge ecod my sicee gatitude to the Uivesity Gat Commissio (UGC) fo havig sactioed a mio eseach poject o the title Fuzzy Diffeetial Equatios This study has eabled me to big out this papei also thak the maagemet of Bishop Hebe College fo thei suppot ecouagemet REFERECES [1] SAbbasbyT AllahVialoo umeical Solutio of fuzzy diffeetial equatios by Taylo method Joual of Computatioal Methods i Applied Mathematics2(2)pp (22) [2] SAbbasbyTAllah Vialoo umeical solutio of fuzzy diffeetial equatios by Ruge-Kutta method oliea studies 11(1)pp (24) [3] JJBuckley EEslami Itoductio to Fuzzy Logic Fuzzy Sets Physica-Velag HeidelbegGemay 21 [4] JJBuckley EEslami TFeuig Fuzzy Mathematics i Ecoomics EgieeigPhysica-Velag Heidelbeg Gemay22 [5] JC Butche The umeical Aalysis of Odiay Diffeetial equatios Ruge-Kutta Geeal Liea Methods ew Yok: Wiley(1987) [6] SLChag LAZadehO Fuzzy Mappig Cotol IEEE Tas Systems Ma Cybeet2 pp3-34(1972) [7] DDuboisHPade Towads fuzzy diffeetial calculus:pat3 Diffeetiatio Fuzzy sets systems 8 pp (1982) [8] CDuaisamy BUsha Aothe appoach to solutio of Fuzzy Diffeetial Equatios Applied Mathematical scieces Vol4o (21) [9] DGoeke Johso Ruge Kutta with highe ode deivative Appoximatios Appliedumeical Mathematics 34 pp27-218(2) [1]RGoetschel WVoxma Elemetay CalculusFuzzy sets systems18pp-31-43(1986) [11] RGethsi shamila & ECHey Amithaaj umeical Solutios of th ode fuzzy iitial value poblems by o-liea tapezoidal method based o logathimic mea with step size cotol Iteatioal Joual of applied Mathematics & Statistical Scieces Vol 3Issue 3 july [12] RGethsi shamila & ECHey Amithaaj umeical Solutios of th ode fuzzy iitial value poblems by fouth ode Ruge-kutta Method based o Cetoidal mea IOSR joual of Mathematics Vol 6Issue 3(May-ju 213)pp [13] RGethsi shamila & ECHey Amithaaj umeical Solutios of fist ode fuzzy iitial value poblems by o-liea tapezoidal fomulae based o vaiety of meas Idia joual of ReseachVol 3Issue-5 May 214 [14] RGethsi shamila & ECHey Amithaaj umeical Solutios of th ode fuzzy iitial value poblems by fouth ode Ruge-kutta method based o Cota-hamoic mea Iteatioal joual o ecet iovatio teds i computig commuicatiosvol 2 Issue:8ISS: [15] OKaleva Fuzzy diffeetial equatios Fuzzy sets systems 24 pp31-317(1987) [16] OKaleva The Cauchy poblem fo Fuzzy diffeetial equatios Fuzzy sets systems 35pp (199) [17] KKaagaaj MSambath umeical solutio of Fuzzy Diffeetial equatios by Thid ode Ruge-Kutta Method Iteatioal joual of Applied Mathematics Computatio Volume2(4)pp [18] MMa MFiedma Kel A umeical solutios of fuzzy diffeetial equatios Fuzzy sets System 15 pp (1999) [19] azeeudi Yaacob Bahom Saugi A ew Fouth-Ode Embedded method based o the Hamoic mea MathematikaJilidhmlpp [313]

9 [2] Vimala SChethupia ew Multi-Step Ruge Kutta Method fo solvig Fuzzy Diffeetial equatios Mathematical Theoy Modelig ISS (Pape)ISS (olie)vol1 o3211 [21]VimalaSaveethaSChethupiya(21) umeical Solutio of Fuzzy Diffeetial Equatios by Ruge-Kutta Method with Highe ode Deivative Appoximatios Poceedigs of the Iteatioal cofeece o Emegig Teds i Mathematics Compute ApplicatiosIdia:MEPCO schlek Egieeig CollegeSivakasi Tamiladupp (ISB: ) [22]PalligkiisSChGPapageogiouFamelisITH Ruge-Kutta methods fo fuzzy diffeetial equatios Applied Mathematics Computatio 29pp97-15(29) [23] RPoalagusamy SSethilkuma A compaiso of Ruge kutta-fouth odes of vaiety of meas embedded meas o Multilaye Raste C simulatio Joual of Theoetical Applied Ifomatio Techology [24] M L Pui D A Ralescu Diffeetials of Fuzzy Fuctios J Math Aal Appl 91 pp (1983) [25] SSeikkala O the Fuzzy iitial value poblem Fuzzy sets systems 24 pp319-33(1987) [314]